Show Explicit Bijection Between Sets (0,1)

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SUMMARY

This discussion focuses on demonstrating an explicit bijection between sets, specifically the open interval (0,1). A bijection is defined as a one-to-one correspondence between two sets. An example provided illustrates a bijection between finite sets A = {0, 1, 2, 3, 4} and B = {8, 9, 10, 11, 12} using the function f(a) = a + 8. The conversation emphasizes the necessity of clearly defining both sets involved in the bijection, as seen in the examples of potential bijections from (0,1) to other intervals.

PREREQUISITES
  • Understanding of bijections in set theory
  • Familiarity with function notation and definitions
  • Knowledge of open intervals in real analysis
  • Basic concepts of finite and infinite sets
NEXT STEPS
  • Explore the concept of bijections in infinite sets, particularly between intervals
  • Learn about Cantor's diagonal argument and its implications for bijections
  • Study the properties of continuous functions and their role in establishing bijections
  • Investigate the use of bijections in real analysis and topology
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Mathematicians, educators, and students studying set theory, real analysis, or anyone interested in understanding the concept of bijections and their applications in mathematics.

jihyel
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How can I show an explicit bijection between sets?
anyidea?
how about with (0,1)?
 
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You just write down a function. Do you know what the definition bijection is?

For example, if you want to show a bijection between A = {0, 1, 2, 3, 4} and B = {8, 9, 10, 11, 12} you could write
[tex]f: A \to B, f(a) = a + 8[/tex]
(for finite sets, you can also write it out as: define f by f(0) = 8, f(1) = 9, ..., f(4) = 12).

how about with (0,1)?
How about it? If you mean: how about a bijection, you are asking half a question. A bijection is always between two sets. "How about a bijection from (0, 1) to (3, 4)", or "from (0, 1) to (0, 1)", or "between (0, 1) and (-2, 2)" would make sense.
 
Compuchip is correct. Think over it.
 

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